Universal algebra Basics of universal algebra: • signatures and algebras • homomorphisms, subalgebras, congruences • equations and varieties • equational calculus • equational specifications and initial algebras • variations: partial algebras, first-order structures Plus some hints on applications in foundations of software semantics, verification, specification, development. . . Andrzej Tarlecki: Semantics & Verification - 215 -
Tiny data type Its signature Σ (syntax): Int , Bool ; sorts 0 , 1: Int ; opns plus , times , minus : Int × Int → Int ; false , true : Bool ; lteq : Int × Int → Bool ; not : Bool → Bool ; and : Bool × Bool → Bool ; and Σ - algebra A (semantics): A Int = Int , A Bool = Bool carriers operations 0 A = 0 , 1 A = 1 plus A ( n, m ) = n + m, times A ( n, m ) = n ∗ m minus A ( n, m ) = n − m false A = ff , true A = tt lteq A ( n, m ) = tt if n ≤ m else ff not A ( b ) = tt if b = ff else ff and A ( b, b ′ ) = tt if b = b ′ = tt else ff Andrzej Tarlecki: Semantics & Verification - 216 -
Signatures Algebraic signature : Σ = ( S, Ω) • sort names : S • operation names, classified by arities and result sorts : Ω = � Ω w,s � w ∈ S ∗ ,s ∈ S Alternatively: Σ = ( S, Ω , arity , sort ) with sort names S , operation names Ω , and arity and result sort functions arity : Ω → S ∗ and sort : Ω → S . • f : s 1 × . . . × s n → s stands for s 1 , . . . , s n , s ∈ S and f ∈ Ω s 1 ...s n ,s Compare the two notions Andrzej Tarlecki: Semantics & Verification - 217 -
Fix a signature Σ = ( S, Ω) for a while. Algebras • Σ -algebra : A = ( | A | , � f A � f ∈ Ω ) • carrier sets: | A | = �| A | s � s ∈ S • operations : f A : | A | s 1 × . . . × | A | s n → | A | s , for f : s 1 × . . . × s n → s • the class of all Σ -algebras: Alg (Σ) Can Alg (Σ) be empty? Finite? Can A ∈ Alg (Σ) have empty carriers? Andrzej Tarlecki: Semantics & Verification - 218 -
Subalgebras • for A ∈ Alg (Σ) , a Σ -subalgebra A sub ⊆ A is given by subset | A sub | ⊆ | A | closed under the operations: − for f : s 1 × . . . × s n → s and a 1 ∈ | A sub | s 1 , . . . , a n ∈ | A sub | s n , f A sub ( a 1 , . . . , a n ) = f A ( a 1 , . . . , a n ) • for A ∈ Alg (Σ) and X ⊆ | A | , the subalgebra of A generated by X , � A � X , is the least subalgebra of A that contains X . • A ∈ Alg (Σ) is reachable if � A � ∅ coincides with A . For any A ∈ Alg (Σ) and X ⊆ | A | , � A � X exists. Fact: Proof (idea): • generate the generated subalgebra from X by closing it under operations in A ; or • the intersection of any family of subalgebras of A is a subalgebra of A . Andrzej Tarlecki: Semantics & Verification - 219 -
Homomorphisms • for A, B ∈ Alg (Σ) , a Σ -homomorphism h : A → B is a function h : | A | → | B | that preserves the operations: − for f : s 1 × . . . × s n → s and a 1 ∈ | A | s 1 , . . . , a n ∈ | A | s n , h s ( f A ( a 1 , . . . , a n )) = f B ( h s 1 ( a 1 ) , . . . , h s n ( a n )) Given a homomorphism h : A → B and subalgebras A sub of A and B sub of B , Fact: the image of A sub under h , h ( A sub ) , is a subalgebra of B , and the coimage of B sub under h , h − 1 ( B sub ) , is a subalgebra of A . Given a homomorphism h : A → B and X ⊆ | A | , h ( � A � X ) = � B � h ( X ) . Fact: Identity function on the carrier of A ∈ Alg (Σ) is a homomorphism Fact: id A : A → A . Composition of homomorphisms h : A → B and g : B → C is a homomorphism h ; g : A → C . Andrzej Tarlecki: Semantics & Verification - 220 -
Isomorphisms • for A, B ∈ Alg (Σ) , a Σ -isomorphism is any Σ -homomorphism i : A → B that has an inverse , i.e., a Σ -homomorphism i − 1 : B → A such that i ; i − 1 = id A and i − 1 ; i = id B . • Σ -algebras are isomorphic if there exists an isomorphism between them. Fact: A Σ -homomorphism is a Σ -isomorphism iff it is bijective (“1-1” and “onto”). Fact: Identities are isomorphisms, and any composition of isomorphisms is an isomorphism. Andrzej Tarlecki: Semantics & Verification - 221 -
Congruences • for A ∈ Alg (Σ) , a Σ -congruence on A is an equivalence ≡ ⊆ | A | × | A | that is closed under the operations: − for f : s 1 × . . . × s n → s and a 1 , a ′ 1 ∈ | A | s 1 , . . . , a n , a ′ n ∈ | A | s n , if a 1 ≡ s 1 a ′ 1 , . . . , a n ≡ s n a ′ n then f A ( a 1 , . . . , a n ) ≡ s f A ( a ′ 1 , . . . , a ′ n ) . For any relation R ⊆ | A | × | A | on the carrier of a Σ -algebra A , there exists Fact: the least congruence on A that contains R . For any Σ -homomorphism h : A → B , the kernel of h , K ( h ) ⊆ | A | × | A | , Fact: where a K ( h ) a ′ iff h ( a ) = h ( a ′ ) , is a Σ -congruence on A . Andrzej Tarlecki: Semantics & Verification - 222 -
Quotients • for A ∈ Alg (Σ) and Σ -congruence ≡ ⊆ | A | × | A | on A , the quotient algebra A/ ≡ is built in the natural way on the equivalence classes of ≡ : − for s ∈ S , | A/ ≡| s = { [ a ] ≡ | a ∈ | A | s } , with [ a ] ≡ = { a ′ ∈ | A | s | a ≡ a ′ } − for f : s 1 × . . . × s n → s and a 1 ∈ | A | s 1 , . . . , a n ∈ | A | s n , f A/ ≡ ([ a 1 ] ≡ , . . . , [ a n ] ≡ ) = [ f A ( a 1 , . . . , a n )] ≡ The above is well-defined; moreover, the natural map that assigns to every Fact: element its equivalence class is a Σ -homomorphisms [ ] ≡ : A → A/ ≡ . Given two Σ -congruences ≡ and ≡ ′ on A , ≡ ⊆ ≡ ′ iff there exists a Fact: Σ -homomorphism h : A/ ≡ → A/ ≡ ′ such that [ ] ≡ ; h = [ ] ≡ ′ . For any Σ -homomorphism h : A → B , A/K ( h ) is isomorphic with h ( A ) . Fact: Andrzej Tarlecki: Semantics & Verification - 223 -
Products • for A i ∈ Alg (Σ) , i ∈ I , the product of � A i � i ∈I , � i ∈I A i is built in the natural way on the Cartesian product of the carriers of A i , i ∈ I : − for s ∈ S , | � i ∈I A i | s = � i ∈I | A i | s − for f : s 1 × . . . × s n → s and a 1 ∈ | � i ∈I A i | s 1 , . . . , a n ∈ | � i ∈I A i | s n , for i ∈ I , f � i ∈I A i ( a 1 , . . . , a n )( i ) = f A i ( a 1 ( i ) , . . . , a n ( i )) For any family � A i � i ∈I of Σ -algebras, projections π i ( a ) = a ( i ) , where i ∈ I Fact: and a ∈ � i ∈I | A i | , are Σ -homomorphisms π i : � i ∈I A i → A i . Define the product of the empty family of Σ -algebras. When the projection π i is an isomorphism? Andrzej Tarlecki: Semantics & Verification - 224 -
Terms Consider an S -sorted set X of variables. • terms t ∈ | T Σ ( X ) | are built using variables X , constants and operations from Ω in the usual way: | T Σ ( X ) | is the least set such that − X ⊆ | T Σ ( X ) | − for f : s 1 × . . . × s n → s and t 1 ∈ | T Σ ( X ) | s 1 , . . . , t n ∈ | T Σ ( X ) | s n , f ( t 1 , . . . , t n ) ∈ | T Σ ( X ) | s • for any Σ -algebra A and valuation v : X → | A | , the value t A [ v ] of a term t ∈ | T Σ ( X ) | in A under v is determined inductively: − x A [ v ] = v s ( x ) , for x ∈ X s , s ∈ S − ( f ( t 1 , . . . , t n )) A [ v ] = f A (( t 1 ) A [ v ] , . . . , ( t n ) A [ v ]) , for f : s 1 × . . . × s n → s and t 1 ∈ | T Σ ( X ) | s 1 , . . . , t n ∈ | T Σ ( X ) | s n Above and in the following: assuming unambiguous “parsing” of terms! Andrzej Tarlecki: Semantics & Verification - 225 -
Term algebras Consider an S -sorted set X of variables. • The term algebra T Σ ( X ) has the set of terms as the carrier and operations defined “syntactically”: − for f : s 1 × . . . × s n → s and t 1 ∈ | T Σ ( X ) | s 1 , . . . , t n ∈ | T Σ ( X ) | s n , f T Σ ( X ) ( t 1 , . . . , t n ) = f ( t 1 , . . . , t n ) . For any S -sorted set X of variables, Σ -algebra A and valuation v : X → | A | , Fact: there is a unique Σ -homomorphism v # : T Σ ( X ) → A that extends v . Moreover, for t ∈ | T Σ ( X ) | , v # ( t ) = t A [ v ] . id X ֒ →| T Σ ( X ) | ✲ | T Σ ( X ) | T Σ ( X ) X ❍❍❍❍❍❍❍❍❍ | v # | Set S Alg (Σ) ∃ ! v # v ❄ ❄ ❥ | A | A Andrzej Tarlecki: Semantics & Verification - 226 -
Equations • Equation : ∀ X.t = t ′ where: − X is a set of variables, and − t, t ′ ∈ | T Σ ( X ) | s are terms of a common sort. • Satisfaction relation : Σ -algebra A satisfies ∀ X.t = t ′ = ∀ X.t = t ′ A | when for all v : X → | A | , t A [ v ] = t ′ A [ v ] . Andrzej Tarlecki: Semantics & Verification - 227 -
Semantic entailment Φ | = Σ ϕ Σ -equation ϕ is a semantic consequence of a set of Σ -equations Φ if ϕ holds in every Σ -algebra that satisfies Φ . BTW: • Models of a set of equations: Mod (Φ) = { A ∈ Alg (Σ) | A | = Φ } • Theory of a class of algebras: Th ( C ) = { ϕ | C | = ϕ } • Φ | = ϕ ⇐ ⇒ ϕ ∈ Th ( Mod (Φ)) • Mod and Th form a Galois connection Andrzej Tarlecki: Semantics & Verification - 228 -
Equational calculus ∀ X.t ′ = t ′′ ∀ X.t = t ′ ∀ X.t = t ′ ∀ X.t ′ = t ∀ X.t = t ′′ ∀ X.t = t ∀ X.t 1 = t ′ ∀ X.t n = t ′ ∀ X.t = t ′ . . . 1 n for θ : X → | T Σ ( Y ) | ∀ X.f ( t 1 . . . t n ) = f ( t ′ 1 . . . t ′ ∀ Y.t [ θ ] = t ′ [ θ ] n ) Mind the variables! a = b does not follow from a = f ( x ) and f ( x ) = b , unless. . . Andrzej Tarlecki: Semantics & Verification - 229 -
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